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Effects of marine sediment on the response of a submerged floating tunnel to P-wave incidence

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Abstract

Submerged floating tunnels (SFTs) are a novel type of traffic structure for crossing long straits or deep lakes. To investigate the dynamic pressure acting on an SFT under compression (P) wave incidence, a theoretical analysis model considering the effect of marine sediment is proposed. Based on displacement potential functions, the reflection and refraction coefficients of P-waves in different media are derived. Numerical examples are employed to illustrate the effects of the thickness of the sediment layer, the incident P-wave angle, the tether stiffness and spacing, and the permeability of the sediment on the dynamic pressure loading on the SFT. The results show that the dynamic pressure is related to the saturation of the sediment and affected by its thickness. Partially saturated sediment will amplify the dynamic pressure loading on the SFT, and the resonance frequency increases slightly with fully saturated sediment. Besides, increasing the tether stiffness or decreasing the tether spacing will decrease the dynamic pressure. Locating the SFT at greater depth and reducing the permeability of the sediment are effective measures to reduce the dynamic pressure acting on the SFT.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 51541810 and 51279178) and the Fundamental Research Funds for the Central Universities (Grant 2018QNA4032).

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Authors and Affiliations

  1. Department of Civil Engineering, College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310058, China

    Heng Lin, Yiqiang Xiang, Zhengyang Chen & Ying Yang

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  1. Heng Lin
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  2. Yiqiang Xiang
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Corresponding author

Correspondence to Yiqiang Xiang.

Appendix A

Appendix A

The elements of the matrix a are listed below:


\( a_{0101} = k_{\mathrm{sP}z} \), \( a_{0102} = k_{x} \), \( a_{0103} = k_{{\mathrm{mP}_{1} z}} \), \( a_{0104} = - k_{{\mathrm{mP}_{1} z}} \), \( a_{0105} = - k_{x} \), \( a_{0106} = k_{{\mathrm{mP}_{2} z}} \), \( a_{0107} = - k_{{\mathrm{mP}_{2} z}} \), \( a_{0108} = - k_{x} \), \( a_{0109} = 0 \), \( a_{0110} = 0 \), \( a_{0111} = 0 \), \( a_{0112} = 0 \); \( a_{0201} = k_{\mathrm{sP}z} \), \( a_{0202} = k_{x} \), \( a_{0203} = \delta_{{\mathrm{P}_{1} }} k_{{\mathrm{mP}_{1} z}} \), \( a_{0204} = - \delta_{{\mathrm{P}_{1} }} k_{{\mathrm{mP}_{1} z}} \), \( a_{0205} = - \delta_{\rm S} k_{x} \), \( a_{0206} = \delta_{{\mathrm{P}_{2} }} k_{{\mathrm{mP}_{2} z}} \), \( a_{0207} = - \delta_{{\mathrm{P}_{2} }} k_{{\mathrm{mP}_{2} z}} \), \( a_{0208} = - \delta_{\rm S} k_{x} \), \( a_{0209} = 0 \), \( a_{0210} = 0 \), \( a_{0211} = 0 \), \( a_{0212} = 0 \); \( a_{0301} = - k_{x} \), \( a_{0302} = k_{\mathrm{sP}z} \), \( a_{0303} = k_{x} \), \( a_{0304} = k_{x} \), \( a_{0305} = k_{\mathrm{mS}z} \), \( a_{0306} = k_{x} \), \( a_{0307} = k_{x} \), \( a_{0308} = - k_{\mathrm{mS}z} \), \( a_{0309} = 0 \), \( a_{0310} = 0 \), \( a_{0311} = 0 \), \( a_{0312} = 0 \); \( a_{0401} = - (\lambda_{\text{s}} {k}_{\rm sp}^{2} { + 2}\mu_{\text{s}} {k}_{\mathrm{sP}z}^{2} ) \), \( a_{0402} = - 2\mu_{\text{s}} {k}_{\mathrm{sS}z} k_{x} \), \( a_{0403} = (A + Q)k_{{\mathrm{mP}_{1} }}^{2} + (Q + R)\delta_{{\mathrm{P}_{{_{1} }} }} k_{{\mathrm{mP}_{1} }}^{2} + 2Gk_{{\mathrm{mP}_{1} z}}^{2} \), \( a_{0404} = ({A} + {Q}){k}_{{\mathrm{mP}_{1} }}^{2} + ({Q} + {R})\delta_{{\mathrm{P}_{1} }} {k}_{{\mathrm{mP}_{1} }}^{2} + 2G{k}_{{\mathrm{mP}_{1} {z}}}^{2} \), \( a_{0405} = - 2Gk_{\mathrm{mS}z} k_{x} \), \( a_{0406} = ({A} + {Q}){k}_{{\mathrm{mP}_{2} }}^{2} + ({Q} + {R})\delta_{{\mathrm{P}_{2} }} {k}_{{\mathrm{mP}_{2} }}^{2} + 2G{k}_{{\mathrm{mP}_{2} {z}}}^{2} \), \( a_{0407} = ({A} + {Q}){k}_{{\mathrm{mP}_{2} }}^{2} + ({Q} + {R})\delta_{{\mathrm{P}_{2} }} {k}_{{\mathrm{mP}_{2} }}^{2} + 2G{k}_{{\mathrm{mP}_{2} {z}}}^{2} \), \( a_{0408} = 2Gk_{\mathrm{mS}z} k_{x} \), \( a_{0409} = 0 \), \( a_{0410} = 0 \), \( a_{0411} = 0 \), \( a_{0412} = 0 \); \( a_{0501} = 2\mu_{\text{s}} k_{\mathrm{sP}z} k_{x} \), \( a_{0502} = (k_{x}^{2} - {k}_{\mathrm{sS}z}^{2} )\mu_{\text{s}} \), \( a_{0503} = 2Gk_{{\mathrm{mP}_{1} z}} k_{x} \), \( a_{0504} = - 2Gk_{{\mathrm{mP}_{1} z}} k_{x} \), \( a_{0505} = - (k_{x}^{2} - {k}_{\mathrm{mS}z}^{2} ){G} \), \( a_{0506} = 2Gk_{{\mathrm{mP}_{2} z}} k_{x} \), \( a_{0507} = - 2Gk_{{\mathrm{mP}_{2} z}} k_{x} \), \( a_{0508} = - (k_{x}^{2} - {k}_{\mathrm{mS}z}^{2} ){G} \), \( a_{0509} = 0 \), \( a_{0510} = 0 \), \( a_{0511} = 0 \), \( a_{0512} = 0 \); \( a_{0601} = 0 \), \( a_{0602} = 0 \), \( a_{0603} = [(1 - {n}) + {n}\delta_{{\mathrm{P}_{1} }} ]{k}_{{\mathrm{mP}_{1} z}} \varphi_{1} ( - {h}_{1} ) \), \( a_{0604} = [(1 - {n}) + {n}\delta_{{\mathrm{P}_{1} }} ]{k}_{{\mathrm{mP}_{1} z}} \varphi_{1} ({h}_{1} ) \), \( a_{0605} = [(1 - {n}) + {n}\delta_{\rm S} ]{k}_{x} \varphi_{2} ( - {h}_{1} ) \), \( a_{0606} = [(1 - {n}) + {n}\delta_{{\mathrm{P}_{2} }} ]{k}_{{\mathrm{mP}_{2} z}} \varphi_{3} ( - {h}_{1} ) \), \( a_{0607} = [(1 - {n}) + {n}\delta_{{\mathrm{P}_{2} }} ]{k}_{{\mathrm{mP}_{2} z}} \varphi_{3} ({h}_{1} ) \), \( a_{0608} = [(1 - {n}) + {n}\delta_{\rm S} ]{k}_{x} \varphi_{2} ({h}_{1} ) \), \( a_{0609} = k_{\mathrm{w}z} \varphi_{4} ( - {h}_{1} ) \), \( a_{0610} = - k_{\mathrm{w}z} \varphi_{4} ({h}_{1} ) \), \( a_{0611} = 0 \), \( a_{0612} = 0 \); \( a_{0701} = 0 \), \( a_{0702} = 0 \), \( a_{0703} = - [({A} + {Q}){k}_{{\mathrm{mP}_{1} }}^{2} + ({Q} + {R})\delta_{{\mathrm{P}_{1} }} {k}_{{\mathrm{mP}_{1} }}^{2} + 2G{k}_{{\mathrm{mP}_{1} {z}}}^{2} ]\varphi_{1} ( - {h}_{1} ) \), \( a_{0704} = - [({A} + {Q}){k}_{{\mathrm{mP}_{1} }}^{2} + ({Q} + {R})\delta_{{\mathrm{P}_{1} }} {k}_{{\mathrm{mP}_{1} }}^{2} + 2G{k}_{{\mathrm{mP}_{1} {z}}}^{2} ]\varphi_{1} ({h}_{1} ) \), \( a_{0705} = 2Gk_{\mathrm{mS}z} k_{x} \varphi_{2} ( - {h}_{1} ) \), \( a_{0706} = - [({A} + {Q}){k}_{{\mathrm{mP}_{2} }}^{2} + ({Q} + {R})\delta_{{\mathrm{P}_{2} }} {k}_{{\mathrm{mP}_{2} }}^{2} + 2G{k}_{{\mathrm{mP}_{2} {z}}}^{2} ]\varphi_{3} ( - {h}_{1} ) \), \( a_{0707} = - [({A} + {Q}){k}_{{\mathrm{mP}_{2} }}^{2} + ({Q} + {R})\delta_{{\mathrm{P}_{2} }} {k}_{{\mathrm{mP}_{2} }}^{2} + 2G{k}_{{\mathrm{mP}_{2} {z}}}^{2} ]\varphi_{3} ({h}_{1} ) \), \( a_{0708} = - 2Gk_{\mathrm{mS}z} k_{x} \varphi_{2} ({h}_{1} ) \), \( a_{0709} = \omega^{2} \rho_{\rm w} \varphi_{4} ( - {h}_{1} ) \), \( a_{0710} = \omega^{2} \rho_{\rm w} \varphi_{4} ({h}_{1} ) \), \( a_{0711} = 0 \), \( a_{0712} = 0 \); \( a_{0801} = 0 \), \( a_{0802} = 0 \), \( a_{0803} = - [({Q} + {R}\delta_{{\mathrm{P}_{1} }} ){k}_{{\mathrm{mP}_{1} }}^{2} ]\varphi_{1} ( - {h}_{1} ) \), \( a_{0804} = - [({Q} + {R}\delta_{{\mathrm{P}_{1} }} ){k}_{{\mathrm{mP}_{1} }}^{2} ]\varphi_{1} ({h}_{1} ) \), \( a_{0805} = 0 \), \( a_{0806} = - [({Q} + {R}\delta_{{\mathrm{P}_{2} }} ){k}_{{\mathrm{mP}_{2} }}^{2} ]\varphi_{3} ( - {h}_{1} ) \), \( a_{0807} = - [({Q} + {R}\delta_{{\mathrm{P}_{2} }} ){k}_{{\mathrm{mP}_{2} }}^{2} ]\varphi_{3} ({h}_{1} ) \), \( a_{0808} = 0 \), \( a_{0809} = n\omega^{2} \rho_{\rm w} \varphi_{4} ( - {h}_{1} ) \), \( a_{0810} = n\omega^{2} \rho_{\rm w} \varphi_{4} ({h}_{1} ) \), \( a_{0811} = 0 \), \( a_{0812} = 0 \); \( a_{0901} = 0 \), \( a_{0902} = 0 \), \( a_{0903} = - 2Gk_{{\mathrm{mP}_{1} z}} k_{x} \varphi_{1} ( - {h}_{1} ) \), \( a_{0904} = 2Gk_{{\mathrm{mP}_{1} z}} k_{x} \varphi_{1} ({h}_{1} ) \), \( a_{0905} = (k_{x}^{2} - {k}_{\mathrm{mS}z}^{2} ){G}\varphi_{2} ( - {h}_{1} ) \), \( a_{0906} = - 2Gk_{{\mathrm{mP}_{2} z}} k_{x} \varphi_{3} ( - {h}_{1} ) \), \( a_{0907} = 2Gk_{{\mathrm{mP}_{2} z}} k_{x} \varphi_{3} ({h}_{1} ) \), \( a_{0908} = (k_{x}^{2} - {k}_{\mathrm{mS}z}^{2} ){G}\varphi_{2} ({h}_{1} ) \), \( a_{0909} = 0 \), \( a_{0910} = 0 \), \( a_{0911} = 0 \), \( a_{0912} = 0 \); \( a_{1001} = 0 \), \( a_{1002} = 0 \), \( a_{1003} = \mathrm{i}\frac{K}{{\alpha_{\text{c}} L}}k_{{\mathrm{mP}_{1} z}} \), \( a_{1004} = - \mathrm{i}\frac{K}{{\alpha_{\text{c}} L}}k_{{\mathrm{mP}_{1} z}} \), \( a_{1005} = \mathrm{i}\frac{K}{{\alpha_{\text{c}} L}}k_{x} \), \( a_{1006} = \mathrm{i}\frac{K}{{\alpha_{\text{c}} L}}k_{mP2z} \), \( a_{1007} = - \mathrm{i}\frac{K}{{\alpha_{\text{c}} L}}k_{{\mathrm{mP}_{2} z}} \), \( a_{1008} = - \mathrm{i}\frac{K}{{\alpha_{\text{c}} L}}k_{x} \), \( a_{1009} = \omega^{2} \rho_{\rm w} \varphi_{4} ( - h_{1} - h_{2} ) - \mathrm{i}\frac{K}{L}k_{\mathrm{w}z} \varphi_{4} ( - h_{1} - h_{2} ) \), \( a_{1010} = \omega^{2} \rho_{\text{w}} \varphi_{4} (h_{1} + h_{2} ) + \mathrm{i}\frac{K}{L}k_{\mathrm{w}z} \varphi_{4} (h_{1} + h_{2} ) \), \( a_{1011} = - \omega^{2} \rho_{\text{w}} \varphi_{5} ( - h_{1} - h_{2} ) \), \( a_{1012} = - \omega^{2} \rho_{\text{w}} \varphi_{5} (h_{1} + h_{2} ) \); \( a_{1101} = 0 \), \( a_{1102} = 0 \), \( a_{1103} = 0 \), \( a_{1104} = 0 \), \( a_{1105} = 0 \), \( a_{1106} = 0 \), \( a_{1107} = 0 \), \( a_{1108} = 0 \), \( a_{1109} = - k_{\mathrm{w}z} \varphi_{4} ( - {h}_{1} - {h}_{2} ) \), \( a_{1110} = k_{\mathrm{w}z} \varphi_{4} ({h}_{1} { + h}_{2} ) \), \( a_{1111} = k_{\text{w}} \varphi_{5} ( - h_{1} - h_{2} ) \), \( a_{1112} = - k_{\text{w}} \varphi_{5} (h_{1} + h_{2} ) \); \( a_{1201} = 0 \), \( a_{1202} = 0 \), \( a_{1203} = 0 \), \( a_{1204} = 0 \), \( a_{1205} = 0 \), \( a_{1206} = 0 \), \( a_{1207} = 0 \), \( a_{1208} = 0 \), \( a_{1209} = 0 \), \( a_{1210} = 0 \), \( a_{1211} = \varphi_{5} ( - {H}) \), \( a_{1212} = \varphi_{5} ({H}) \); with \( \varphi_{1} (x) = \exp (\mathrm{i}k_{{\mathrm{mP}_{1} z}} x) \), \( \varphi_{2} (x) = \exp (\mathrm{i}k_{\mathrm{mS}z} x) \), \( \varphi_{3} (x) = \exp (\mathrm{i}k_{{\mathrm{mP}_{2} z}} x) \), \( \varphi_{4} (x) = \exp (\mathrm{i}k_{\mathrm{w}z} x) \), \( \varphi_{5} (x) = \exp (\mathrm{i}k_{\rm w} x) \).

The elements of the matrix f are listed below:

\( f_{01} = k_{\mathrm{sP}z} \), \( f_{02} = k_{\mathrm{sP}z} \), \( f_{03} = k_{x} \), \( f_{04} = \lambda_{\text{s}} k_{\rm sp}^{2} + 2\mu_{\text{s}} k_{\mathrm{sP}z}^{2} \), \( f_{05} = 2\mu_{\text{s}} k_{\mathrm{sP}z} k_{x} \), \( f_{06} = 0 \), \( f_{07} = 0 \), \( f_{08} = 0 \), \( f_{09} = 0 \), \( f_{10} = 0 \), \( f_{11} = 0 \), \( f_{12} = 0 \).

The elements of the matrix aʹ are listed below:

\( a^{{\prime }}_{0101} = k_{\mathrm{sP}z} \), \( a^{{\prime }}_{0102} = k_{x} \), \( a^{{\prime }}_{0103} = k_{\mathrm{w}z} \), \( a^{{\prime }}_{0104} = - k_{\mathrm{w}z} \), \( a^{{\prime }}_{0105} = 0 \), \( a^{{\prime }}_{0106} = 0 \); \( a^{\prime }_{0201} = - (\lambda_{\text{s}} k_{\rm sp}^{2} + 2\mu_{\text{s}} k_{\mathrm{sP}z}^{2} ) \), \( a^{{\prime }}_{0202} = - 2\mu_{\text{s}} k_{\mathrm{sP}z} k_{x} \), \( a^{\prime }_{0203} = \omega^{2} \rho_{\text{w}} \), \( a^{{\prime }}_{0204} = \omega^{2} \rho_{\text{w}} \), \( a^{{\prime }}_{0205} = 0 \), \( a^{{\prime }}_{0206} = 0 \); \( a^{{\prime }}_{0301} = 2k_{\mathrm{sP}z} k_{x} \mu_{\text{s}} \), \( a^{{\prime }}_{0302} = k_{x}^{2} - k_{{sS{\text{z}}}}^{2} \), \( a^{{\prime }}_{0303} = 0 \), \( a^{{\prime }}_{0304} = 0 \), \( a^{{\prime }}_{0305} = 0 \), \( a^{{\prime }}_{0306} = 0 \); \( a^{{\prime }}_{0401} = 0 \), \( a^{{\prime }}_{0402} = 0 \), \( a^{{\prime }}_{0403} = \omega^{2} \rho_{\text{w}} \varphi_{4} ( - h_{1} - h_{2} ) - \mathrm{i}\frac{K}{L}k_{\mathrm{w}z} [\varphi_{4} ( - h_{1} - h_{2} ) - 1] \), \( a^{\prime }_{0404} = \omega^{2} \rho_{\text{w}} \varphi_{4} (h_{1} + h_{2} ) + \mathrm{i}\frac{K}{L}k_{\mathrm{w}z} [\varphi_{4} (h_{1} + h_{2} ) - 1] \), \( a^{{\prime }}_{0405} = - \omega^{2} \rho_{\text{w}} \varphi_{5} ( - h_{1} - h_{2} ) \), \( a^{{\prime }}_{0406} = - \omega^{2} \rho_{\text{w}} \varphi_{5} (h_{1} + h_{2} ) \); \( a^{{\prime }}_{0501} = 0 \), \( a^{{\prime }}_{0502} = 0 \), \( a^{\prime }_{0503} = - k_{\mathrm{w}z} \varphi_{4} ( - h_{1} - h_{2} ) \), \( a^{\prime }_{0504} = k_{\mathrm{w}z} \varphi_{4} (h_{1} + h_{2} ) \), \( a^{{\prime }}_{0505} = k_{\text{w}} \varphi_{4} ( - h_{1} - h_{2} ) \), \( a^{{\prime }}_{0506} = - k_{\text{w}} \varphi_{4} ({h}_{1} { + h}_{2} ) \); \( a^{{\prime }}_{0601} = 0 \), \( a^{{\prime }}_{0602} = 0 \), \( a^{{\prime }}_{0603} = 0 \), \( a^{{\prime }}_{0604} = 0 \), \( a^{\prime }_{0605} = \varphi_{5} ( - H) \), \( a^{{\prime }}_{0606} = \varphi_{5} ({H}) \).

The elements of the matrix fʹ are listed below:

\( f^{{\prime }}_{01} = k_{\mathrm{sP}z} \), \( f^{{\prime }}_{02} = \lambda_{\text{s}} k_{\rm sp}^{2} + 2\mu_{\text{s}} k_{\mathrm{sP}z}^{2} \), \( f^{{\prime }}_{03} = 2\mu_{\text{s}} k_{\mathrm{sP}z} k_{x} \), \( f^{{\prime }}_{04} = 0 \), \( f^{{\prime }}_{05} = 0 \), \( f^{{\prime }}_{06} = 0 \).

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Lin, H., Xiang, Y., Chen, Z. et al. Effects of marine sediment on the response of a submerged floating tunnel to P-wave incidence. Acta Mech. Sin. 35, 773–785 (2019). https://doi.org/10.1007/s10409-019-00847-0

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  • DOI: https://doi.org/10.1007/s10409-019-00847-0

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